Selection of prior distributions and multiscale analysis in Bayesian temperature reconstructions based on fossil assemblages

It is possible to reconstruct the past variation of an environmental variable from measured historical indicators when the modern values of the variable and the indicators are known. In a Bayesian statistical approach, the selection of a prior probability distribution for the past values of the environmental variable can then be crucial and the selection therefore should be made carefully. This is particularly the case when the data are noisy and the statistical model used is complex since the influence of the prior on the results can then be especially strong. It can be difficult to elicit the prior probability distribution from the available information, since usually there are no measured data on the past values of the variable one wants to reconstruct and different reconstructions are typically consistent with each other only at a coarse level. To overcome these difficulties we propose to use a non-informative smoothing prior, possibly in combination with an informative prior, that simply penalizes for roughness of the reconstruction as measured by the variability of its values. We believe that it can sometimes be easier to set an overall prior distribution on the roughness than to agree on a prior for the actual values of the reconstructed variable. Note that by using a smoothing prior one incorporates into the model itself the smoothing step usually done before or after the actual numerical reconstruction. Another idea proposed in this paper is to integrate the reconstruction model with a multiscale feature analysis technique known as SiZer. Multiscale analysis of the posterior distribution of the reconstructed variable makes it possible to infer its statistically significant features such as trends, maxima and minima at several different time scales. While only temperature is considered in this paper, the technique can be applied to other environmental variables.